A Matrix Expander Chernoff Bound

TL;DR

This paper establishes a Chernoff-type bound for matrix-valued sums sampled via expander walks, advancing concentration inequalities in matrix analysis and random walk theory.

Contribution

It introduces a new multi-matrix Golden-Thompson inequality and proves a matrix Chernoff bound for expander walk sampling, confirming a conjecture by Wigderson and Xiao.

Findings

Proved a matrix Chernoff bound for expander walk sampling

Developed a new multi-matrix Golden-Thompson inequality

Showed a reduction from vector martingale concentration to expander walk concentration

Abstract

We prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a random walk on an expander, confirming a conjecture due to Wigderson and Xiao. Our proof is based on a new multi-matrix extension of the Golden-Thompson inequality which improves in some ways the inequality of Sutter, Berta, and Tomamichel, and may be of independent interest, as well as an adaptation of an argument for the scalar case due to Healy. Secondarily, we also provide a generic reduction showing that any concentration inequality for vector-valued martingales implies a concentration inequality for the corresponding expander walk, with a weakening of parameters proportional to the squared mixing time.